Hilbert series of PI relatively free G-graded algebras are rational

arXiv:1011.3253v1 [math.RA] 14 Nov 2010

HILBERT SERIES OF PI RELATIVE FREE G-GRADED ALGEBRAS ARE RATIONAL FUNCTIONS ELI ALJADEFF AND ALEXEI KANEL-BELOV Abstract. Let G be a finite group, (g1 , . . . , gr ) an (unordered) r-tuple of G(r) and xi,gi ’s variables that correspond to the gi ’s, i = 1, . . . , r. Let F hx1,g1 , . . . , xr,gr i be the corresponding free G-graded algebra where F is a field of zero characteristic. Here the degree of a monomial is determined by the product of the indices in G. Let I be a G-graded T -ideal of F hx1,g1 , . . . , xr,gr i which is PI (e.g. any ideal of identities of a G-graded finite dimensional algebra is of this type). We prove that the Hilbert series of F hx1,g1 , . . . , xr,gr i/I is a rational function. More generally, we show that the Hilbert series which corresponds to any g-homogeneous component of F hx1,g1 , . . . , xr,gr i/I is a rational function.

1. Introduction The Hilbert series of an affine (i.e. finitely generated) algebra and its computation, is a topic which attracted a lot of attention in the last century , classically in commutative algebra (see e.g. [29], [30]), but also (and in fact more importantly for the purpose of this paper) in non commutative algebra (see e.g. [5]). In particular the question of when the Hilbert series HW of and algebra W is the Taylor expansion of a rational function is fundamental in the theory, and when it does, it serves as a “fine” structural invariant of W with applications to growth invariants (see [13], [14], [15], [25], [26], [28]). In case the algebra W is a relatively free algebra, i.e. isomorphic to the quotient of an affine free algebra F hx1 , . . . , xn i by a T -ideal of identities I it is known that HF hx1 ,...,xn i/I is a rational function (see [8], [20]) and this fact has been successfully used in the estimation of the asymptotic behavior of the co-character sequence of a PI algebra. Specifically in [11] (based on explicit formulas for the co-character sequences which appear in [10]) the authors show that if A is PI-algebra with 1 which satisfies a Capelli identity, then the asymptotic behavior of the codimension sequence is of the form Date: Oct. 30, 2010. 2000 Mathematics Subject Classification. 16R10, 16W50, 16P90. Key words and phrases. graded algebra, polynomial identity, Hilbert series. The first author was partially supported by the ISRAEL SCIENCE FOUNDATION (grant No. 1283/08) and by the E.SCHAVER RESEARCH FUND. The second author was partially supported by the ISRAEL SCIENCE FOUNDATION (grant No. 1178/06). The second author is grateful to the Russian Fund of Fundamental Research for supporting his visit to India in 2008 (grant RF BR08 − 01 − 91300 − IN Da ). 1

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ELI ALJADEFF AND ALEXEI KANEL-BELOV

cn (A) = ang ln where a is a scalar, 2g is an integer and l is a non negative integer (we refer the reader to [17], [18] and [19] for a comprehensive account on the codimension sequence of a PI algebra). The rationality of the Hilbert series of an affine relatively free algebra has been established also in case W is a super algebra (i.e. Z2 -graded) and our goal in this paper is to extend these results to G-graded relatively free affine PI algebras where G is an arbitrary finite group. Let Ω = (g1 , ..., gr ) be an unordered r-tuple of elements of G (in particular we allow repetitions). Let XG = {x(1,g1 ) , ..., x(r,gr ) } be a set of variables which correspond to the set Ω and let F hXG i be the free algebra generated by XG over F . In order to keep the notation as light as possible we may omit one index and write XG = {xg1 , ..., xgr } with the convention that the variables xgi and xgj are different even if gi = gj in G. We equip F hXG i with a (natural) G-grading, namely a monomial of the form xgs1 xgs2 · · · xgsm is homogeneous of degree gs1 gs2 · · · gsm ∈ G. We also consider its homogeneous degree m ∈ N, namely the number of variables in the monomial. Let I be a G-graded T -ideal, that is, I is closed under G-graded endomorphisms of F hXG i. This implies (by a standard argument) that I is generated by strongly G-graded polynomials, i.e. G-graded polynomials whose different monomials are permutations of each other and moreover they all have the same G-degree (see [2]). We will assume in addition that the T -ideal I contains an ordinary polynomial. By this we mean that I contains all G-graded polynomials obtained by assigning all possible degrees in G to the variables xi of an ordinary non zero polynomial p(x1 , . . . , xn ). We refer to a G-graded T -ideal which contains an ordinary polynomial as a PI G-graded T -ideal. Remark 1.1. Note for instance, that the T -ideal of G-graded identities of any finite dimensional G-graded algebra is PI. On the other hand if G 6= {e}, then the G-graded algebra W where We is a free noncommutative algebra and Wg = 0 for g 6= e is G-graded PI but of course not PI. Let F hxg1 , . . . , xgr i be the free G-graded algebra generated by homogeneous variables {xgi }ri=1 . As above we let I be a G-graded T -ideal which is PI and let Φ = F hxg1 , . . . , xgr i/I be the corresponding relatively free G-graded algebra. Let Ωn be the (finite) set of monomials of degree n on the xg ’s and let cn be the dimension of the F -subspace of Φ spanned by the monomials of Ωn . We denote by X HF hxg1 ,...,xgr i/I (t) = c n tn n

the Hilbert series of Φ with respect to the xg ’s. Theorem 1.2. The series HF hxg1 ,...,xgr i/I (t) is the Taylor series of a rational function. In our proof we strongly use key ingredients which appear in the proof of representability of the G-graded relatively free algebras (see [3]). These ingredients include the existence of certain polynomials, called Kemer polynomials, which are “extremal non identities” (see [3]) and their existence relies on the fundamental fact that the Jacobson radical of a PI algebra is nilpotent (see [4], [12], [22] and

HILBERT SERIES OF PI RELATIVE FREE G-GRADED ALGEBRAS ARE RATIONAL FUNCTIONS 3

[27]) (this parallels the Hilbert’s Nullstellensatz in the commutative theory), and the solution of the Specht problem (see [3], [23], [24] and [31]) (which parallels the Hilbert basis theorem in the commutative theory). We emphasize however that the rationality of the Hilbert series is not a corollary of representability since there are examples of representable algebras which have a transcendental Hilbert series. Indeed, recall from [20] that the monomial algebra supported by monomials of the n 2 2 form h = x1 xm 2 x3 x4 x5 with the extra relation that h = 0 if m − 2n = 1 is representable but with a transcendental Hilbert series. For more on this we refer the reader to [9] and [21]. In Theorems 1.5 and 1.8 below we present different generalizations of Theorem 1.2. Then, in Theorem 1.9, these are combined into one general statement. We start with multivariate Hilbert series. Let F hxg1 , . . . , xgr i be the free G-graded algebra generated by homogeneous variables {xgi }ri=1 . As above we let I be a G-graded T -ideal which is PI and let F hxg1 , . . . , xgr i/I be the corresponding relatively free G-graded algebra. For any rtuple of non-negative integers (d1 , . . . , dr ) we consider the (finite) set of monomials Ω(d1 ,...,dr ) on xg ’s where the variable xgi appears exactly di times, i = 1, . . . , r. We denote by c(d1 ,...,dr ) the dimension of the F -subspace of F hxg1 , . . . , xgr i/I spanned by the monomials in Ω(d1 ,...,dr ) . Remark 1.3. As mentioned above the T ideal I is generated by multilinear Ggraded polynomials which are strongly homogeneous i.e. its monomials have the same homogeneous degree in G. This fact will play an important role in the sequel. Definition 1.4. Notation as above. The multivariate Hilbert series of F hxg1 , . . . , xgr i/I is given by P HF hxg1 ,...,xgr i/I (t1 , . . . , tr ) = (d1 ,...,dr ) c(d1 ,...,dr ) td11 · · · tdr r . The following result generalizes Theorem 1.2. Theorem 1.5. Notation as above. The Hilbert series HF hxg1 ,...,xgr i/I (t1 , . . . , tr ) is a rational function. Remark 1.6. Clearly, Theorem 1.2 follows from Theorem 1.5 simply by replacing all variables ti by a single variable t. The next generalization of Theorem 1.2 is in a different direction. We consider the Hilbert series (in one variable t) of a unique homogeneous component. More precisely we fix g ∈ G and we consider the set of monomials Ωg,n of degree n whose homogeneous degree is g. We let cg,n be the dimension of the subspace in F hxg1 , . . . , xgr i/I spanned by the monomials in Ωg,n (or rather, by the elements in F hxg1 , . . . , xgr i/I they represent). Definition 1.7. With the above notation, the Hilbert series of the g-component of F hxg1 , . . . , xgr i/I is given by X cg,n tn Hg,F hxg1 ,...,xgr i/I (t) = Theorem 1.8. With the above notation. The Hilbert series Hg,F hxg1 ,...,xgr i/I (t) is a rational function. The case where g = e is of particular interest. Moreover we could consider the Hilbert series of any collection of g-homogeneous components and in particular the Hilbert series which corresponds to a subgroup H of G.

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Finally we combine the generalizations which appeared in Theorems 1.5 and 1.8. Fix an element g ∈ G. For any r-tuple (d1 , ..., dr ) of non-negative integers we consider the monomials with xg1 appearing d1 times, xg2 appearing d2 times, . . ., whose homogeneous degree is g ∈ G. We denote this set of monomials by Ωg,(d1 ,...,dr ) and we let cg,(d1 ,...,dr ) be the dimension of the space in F hxg1 , . . . , xgr i/I spanned by elements whose representatives are the monomials in Ωg,(d1 ,...,dr ) . Theorem 1.9. The Hilbert series Hg,F hxg1 ,...,xgr i/I (t1 , . . . , tr ) =

X

cg,(d1 ,...,dr ) td11 · · · tdr r

which corresponds to the g-component of the G-graded algebra F hxg1 , . . . , xgr i/I is a rational function. Remark 1.10. Clearly, Theorems 1.2, 1.5, 1.8 are direct corollaries of Theorem 1.9. Theorem 1.9 is proved in Section 2. As mentioned above the proof uses ingredients from the proof of the representability of relatively free G-graded, affine, PI algebras. For the reader convenience we recall in the first part of the section the required results from [3] which are used in the proof. In Section 3 we consider the T -ideal of G-graded identities I of the group algebra F G and show, in a rather direct way, the rationality of the Hilbert series of the corresponding relatively free algebra F hxg1 , . . . , xgr i/I. We also give in this case (see Corollary 3.4) the asymptotic behavior of the corresponding codimension sequence. Furthermore, these results can be easily extended to twisted group algebras F α G, where α ∈ Z 2 (G, F ∗ ) (see Remark 3.5). The G-grading determined by twisted group algebras is called “fine” and it plays an important role in the classification of G-graded simple algebras (see [6]). We close the introduction by mentioning that one may consider the Hilbert series of an affine G-graded free algebra. It is easy to see that the corresponding Hilbert series is a rational function. Of course one may ask whether the different generalizations (Theorems 1.5 and 1.8) apply also in this case (i.e. when I = 0). It is easy to see that this is indeed the case when extending to multivariate series. As for the second generalization, we refer the reader to [16]. The authors prove the rationality of the Hilbert series which corresponds to the sub algebra of coinvariant elements, that is, in case g = e. 2. Preliminaries and proofs We start this section by recalling some facts on G-graded algebras W over a field of characteristic zero F and their corresponding G-graded identities. We refer the reader to [3] for a detailed account on this topic. Let W be an affine G-graded PI algebra over F . We denote by I = idG (W ) the ideal of G-graded identities of W . These are polynomials in the free G-graded algebra over F which are generated by XG and that vanish upon any admissible S evaluation on W . Here XG = Xg and Xg is a set of countably many variables of degree g. An evaluation is admissible if the variables from Xg are replaced only by elements from Wg . It is known that I is a G-graded T -ideal, i.e. closed under G-graded endomorphisms of F hXG i. We recall from [3] that the T -ideal I is generated by multilinear polynomials. Consequently, the T -ideal of identities does not change when passing to F , the


algebraic closure of F , in the sense that the ideal of identities of WF over F is the span (over F ) of the T -ideal of identities of W over F . It is easily checked that the Hilbert series remains the same when passing to the algebraic closure of F . Thus, from now on we assume F = F . Next, we recall some terminology and some facts from Kemer theory extended to the context of G-graded algebras as they appear in [3]. We start with the concept of alternating polynomial on a set of variables. Let f (x1,g , . . . , xr,g ; y1 , . . . , yn ) be a multilinear polynomial with variables (x1,g , . . . , xr,g ), homogeneous of degree g, and some other variables y’s, homogeneous of unspecified degrees. We say that f is alternating on the set x1,g , . . . , xr,g if there is a multilinear polynomial h(x1,g , . . . , xr,g ; y1 , . . . , yn ) such that f (x1,g , . . . , xr,g ; y1 , . . . , yn ) =

X

(−1)σ h(xσ(1),g , . . . , xσ(r),g ; y1 , . . . , yn ).

σ∈Sym(r)

We say that a polynomial f alternates on a collection of disjoint sets of homogeneous variables (each set constitute of variables of the same degree), if it is alternating on each set. In the sequel we will need to consider multilinear polynomials f which alternates on d disjoint sets of g-elements, each of cardinality r. More generally, we will consider multilinear polynomials such that for any g ∈ G, f contains ng disjoint sets of variables of homogeneous degree g and each set of cardinality dg . We recall from [3] that a G-graded polynomial with an alternating sets of ghomogeneous variables which is “large enough” is necessarily an identity. More precisely, for any affine PI G-graded algebra W and for any g ∈ G there exists an integer dg such that any G-graded polynomial which has an alternating set of g variables of cardinality exceeding dg is necessarily a G-graded identity of W . In particular this holds for a finite dimensional G-graded algebra A. Note that in that case, if a polynomial f has an alternating set of g-homogeneous elements whose cardinality exceeds the dimension of Ag , it is clearly an identity of A. Next we recall that by Wedderburn-Malcev decomposition theorem, a G-graded finite dimensional algebra A over F , may be decomposed into the direct sum of A ⊕ J (decomposition as vector spaces) where J is the Jacobson radical (G-graded) and A is a (semisimple) subalgebra of A isomorphic to A/J as G-graded algebras. As a consequence we have decompositions of A and J to the corresponding ghomogeneous components. Now, we know that in order to test whether a multilinear polynomial is an identity of an algebra, it is sufficient to evaluate its variables on a base and hence, applying the above decomposition, we may consider semisimple and radical Ggraded evaluations. From these considerations we conclude that if a polynomial has sufficiently many alternating sets of g-homogeneous elements of cardinality that exceeds the dimension of the g-homogeneous component of A, the polynomial is necessary an identity of A. Indeed, in any evaluation we either have a semisimple basis element which appears twice or at least one of the evaluations is radical. In the first case we get zero as a result of the alternation of two elements which are equal, while in the second, we get zero if the number of alternating sets is at least the nilpotency index of J. We therefore see that if a G-graded polynomial f has a number of alternating sets (same cardinality) of g-variables which is at least the nilpotency index of J (and in particular if it has “sufficiently many”) then the

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cardinality of the sets must be bounded by the dimension of Ag if we know that f is a non (G-graded) identity of A. Thus, if f is a non (G-graded) identity of A with αg disjoint alternating sets of g-homogeneous elements of cardinality dg + 1, g ∈ G P where dg = dim(Ag ) then, g αg ≤ n − 1, where n is the nilpotency index of J. In [3] the notion of a finite dimensional G-graded basic algebra was introduced. For our exposition here, it is not necessary to recall its precise definition but only say (as a result of Kemer’s Lemmas 1 and 2 for G-graded algebras (see sections 5 and 6 in [3])) that any basic algebra A admits non-identities G-graded polynomials which have “arbitrary many” (say at least n, the nilpotency index of J) alternating sets of g-homogeneous variables of cardinality dg and precisely (a total of) n − 1 alternating sets of g-homogeneous variables of cardinality dg + 1 for some g ∈ G. These extremal non identities are the so called “G-graded Kemer polynomials” of the basic algebra A. Thus to each basic algebra A corresponds an r + 1-tuple of non-negative integers (dg1 , . . . , dgr ; n − 1) where dg is the dimension of the g-component of the semisimple part of A and n is the nilpotency index of J, the radical of A. We refer to such a tuple as the Kemer point of the basic algebra A. The representability theorem for affine G-graded algebras can be stated as follows. Given an affine PI G-graded algebra W over F , where F is a field of zero characteristic, there exists a finite number of G-graded basic algebras A1 , . . . , Am over a field extension K of F such that W satisfies the same T G-graded identities as A1 ⊕ · · · ⊕ Am . Note that since idG (A1 ⊕ · · · ⊕ An ) = idG (Ai ) we may assume id(Ai ) * id(Aj ) for every 1 ≤ i, j ≤ m. Remark 2.1. In fact, by passing to the algebraic closure of K, we may assume that the algebras Ai above are finite dimensional over the same field F . Definition 2.2. With the above notation, we say that a finite dimensional Ggraded algebra A is subdirectly irreducible if it has no non trivial, two sided Ggraded ideals I and J such that I ∩ J = (0). Remark 2.3. Note that if the algebra A1 ⊕ · · ·⊕ Am is subdirectly irreducible then m = 1. A key ingredient in the proof of Theorem 1.9 is the existence of an essential Shirshov base for the relatively free algebra F hxg1 , . . . , xgr i/I. For the reader convenience we recall the necessary definitions and statements from [3], starting from the ordinary case (i.e. ungraded). Definition 2.4. Let W be an affine PI-algebra over F . Let {a1 , . . . , as } be a set of generators of W . Let m be a positive integer and let Y be the set of all words in {a1 , . . . , as } of length ≤ m. We say that W has Shirshov base of length m and of height h if elements of the form yik11 · · · yikll where yii ∈ Y and l ≤ h, span W as a vector space over F . Theorem 2.5. If W is an affine PI-algebra, then it has a Shirshov base for some m and h. More precisely, suppose W is generated by a set of elements of cardinality s and suppose it has PI-degree m (i.e. there exists an identity of degree m and m is minimal) then W has a Shirshov base of length m and of height h where h = h(m, s).


In fact we will need a weaker condition (see [3]) Definition 2.6. Let W be an affine PI-algebra. We say that a set Y as above is an essential Shirshov base of W (of length m and of height h) if there exists a finite set D(W ) such that the elements of the form di1 yik11 di2 · · · dil yikll dil+1 where dij ∈ D(W ), yij ∈ Y and l ≤ h span W . An essential Shirshov’s base gives the following. Theorem 2.7. Let C be a commutative ring and let W = Ch{a1 , . . . , as }i be an affine algebra over C. If W has an essential Shirshov base (in particular, if W has a Shirshov base) whose elements are integral over C, then it is a finite module over C. Returning to G graded algebras we have the following. Proposition 2.8. Let W be an affine, PI, G-graded algebra. Then it has an essential G-graded Shirshov base of elements of We . For the proof of Theorem 1.9 we will assume below there exists a G-graded T -ideal I which is non-rational (that is the Hilbert series of F hxg1 , . . . , xgr i/I is non-rational) and get a contradiction. The next two lemmas will be used to reduce the problem to the case where I is maximal with respect to being non-rational and also that the relatively free algebra F hxg1 , . . . , xgr i/I is G-graded PI equivalent to one basic algebra (rather than to a direct sum of them). Before stating the lemmas we simplify the terminology as follows. Remark 2.9. If X is a subspace of an algebra U = F hxg1 , ..., xgr i/I spanned by strongly homogeneous polynomials, then we may consider naturally its corresponding multivariate Hilbert series. Then when we say “Hilbert series of X” we mean “multivariate G-graded Hilbert series which corresponds to the g-component of X” for any given g ∈ G. Lemma 2.10. Let J be a G-graded T -ideal containing I. Let HU , HU/J and HJ/I be the Hilbert series of U , U/J and J/I respectively. Then HF hxg1 ,...,xgr i/I = HF hxg1 ,...,xgr i/J + HJ/I . Proof. This is clear since J is spanned by strongly homogeneous polynomials. ′

Lemma 2.11. Let I and I ′′ be two G-graded T -ideals which contain I. Then the following holds: HF hxg

1 ,...,xgr i/(I

′

∩I ′′ )

= HF hxg

1 ,...,xgr i/I

′

+ HF hxg

1 ,...,xgr i/I

′′

− HF hxg

1 ,...,xgr i/(I

′

+I ′′ )

Proof. The proof is similar to the proof of Lemma 9.40 in [20] and hence is omitted. Let us assume now that there exist G-graded T -ideals of F hxg1 , ..., xgr i which are PI and such that the Hilbert series of F hxg1 , ..., xgr i/I is non-rational. Proposition 2.12. Under the above assumption there exists a G-graded T -ideal of F hxg1 , ..., xgr i which is PI and

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(1) is maximal with respect to the property that the Hilbert series of F hxg1 , ..., xgr i/I is non-rational. (2) the relatively free algebra F hxg1 , ..., xgr i/I is G-graded PI equivalent to a G-graded basic algebra A. Proof. The first assertion follows from the G-graded Specht property (see section 12 in [3]). Indeed, if there is no such an ideal then we get an infinite ascending sequence of ideals which does not stabilize and this contradicts the fact that the union of the T -ideals is finitely generated. We thus may assume that the Hilbert series of F hxg1 , ..., xgr i/I is non-rational and the Hilbert series of F hxg1 , ..., xgr i/J for any G-graded T -ideal J which properly contains I is a rational function. For the proof of the second assertion let us show that the maximality of I already implies that F hxg1 , ..., xgr i/I is PI equivalent to a G-graded basic algebra. Assuming the converse, we have that F hxg1 , ..., xgr i/I is G-graded PI equivalent to a direct sum of basic algebras A1 ⊕ A2 ⊕ . . . ⊕ Am where m ≥ 2 and idG (Ai ) * idG (Aj ) for any 1 ≤ i, j ≤ T m. It follows that the ideal idG (F hxg1 , ..., xgr i/I) = idG (A1 ⊕ A2 ⊕ . . . ⊕ Am ) = idG (Ai ) and idG (F hxg1 , ..., xgr i/I) idG (Ai ) for any i. Now, consider the evaluations Ii of the T -ideals idG (Ai ) on F hxg1 , ..., xgr i. Clearly, Ii properly contains I and their intersection is I. By Lemma 2.11 we conclude that the Hilbert series of F hxg1 , ..., xgr i/I is rational. Contradiction. It is convenient to view the relatively free algebra F hxg1 , ..., xgr i/I as an algebra of generic elements, G-graded embedded in a matrix algebra over a suitable rational function field over F . Indeed, by part (2) of Proposition 2.12 we have that F hxg1 , ..., xgr i/I is G-graded PI equivalent to a G-graded basic algebra which we denote by A and which from now on will be viewed a G-graded subalgebra of the n × n matrices over F (see [3]). If {vg,1 , . . . , vg,sg } is an F -basis of Ag , the ghomogeneous component of A, we consider different sets of central indeterminates {tg,1 , . . . , tg,sg }, g ∈ G (one set for each generator xg of the free algebra). Let K = F h{tg,i }i be the field of rational functions on the t’s and AK be the algebra over K obtained from A by extending scalars from F to K. The algebra of generic elements will be an F -subalgebra of AK . For each variable xg = xgj P in the generating set of F hxg1 , ..., xgr i we form an element zg = i tg,i vg,i in AK . Following the embedding of A in Mn (F ) we view the generating elements zg ’s as n×n-matrices over the field K. Observe that these generating elements are matrices whose entries are homogeneous polynomials (on the t’s) of degree one. Proposition 2.13 (see [3]). There is an F -isomorphism of G-graded algebras of F hxg1 , ..., xgr i/I with A, the F -sub algebra of AK generated by the elements {zg1 , . . . , zgr }. Remark 2.14. In the rest of the proof we use without further notice the identification of the relatively free algebra with the algebra of generic elements A embedded in Mn (K). Now we recall from Proposition 2.8 that the relatively free algebra F hxg1 , ..., xgr i/I has an essential Shirshov Θ base which is contained in the e-component. Picking the natural set of generators of F hxg1 , ..., xgr i/I we note that Θ consists of monomials on the xgi ’s and hence, viewed in A, they consist of matrices over K whose entries are homogeneous polynomial in the ti ’s. It then follows that the characteristic values of the elements in Θ are homogeneous polynomials on the ti ’s.


Denote by C the algebra over F generated by these homogeneous polynomials. Note that C is an affine commutative algebra (the essential Shirshov base is finite). Moreover, if we extend the algebra of generic elements A to C we have (by Theorem 2.7) that AC is a finite module over C. For AC we know the following result. Lemma 2.15. The Hilbert series of AC is rational. More generally, the rationality of the Hilbert series is independent of the integer degree given to the generators. Proof. Indeed, the algebra is a finitely generated module over an affine domain and hence its Hilbert series is rational. (see [20], Prop. 9.33). More generally, we may consider C-submodules M of AC which are generated by strongly homogeneous polynomials on the xgi ’s. Lemma 2.16. The Hilbert series of M is rational. Proof. Since AC is a finite module over a Noetherian domain, the module M is finitely generated as well. The result now follows from the second part of Lemma 2.15. We can complete now the proof of Theorem 1.9. Let f be a G-graded Kemer polynomial of the basic algebra A and let J be the G-graded T -ideal it generates together with I. Note that since f is a non identity of A (and hence a non identity of F hxg1 , ..., xgr i/I) the T -ideal J strictly contains I, and hence, by the maximality of I we have that the Hilbert series of F hxg1 , ..., xgr i/J is rational. The key property which we need here is that the ideal J/I is closed under the multiplication of the characteristic values (see [3], Prop. 8.2). Hence the ideal J/I of the relatively free algebra is in fact a C-submodule of AC . Hence its Hilbert series is rational. Applying Lemma 2.10 the result follows. This completes the proof of Theorem 1.9. 3. A special case In this section we show by direct computations the rationality of the Hilbert series of the affine relative free G-graded algebra in case I is the T -ideal of G-graded identities of the group algebra F G. In addition we obtain a precise estimation of the asymptotic behavior of the G-graded codimension sequence for that case. Let α = (g1 , g2 , ..., gr ) be an r-tuple in G(r) . As in previous sections we consider the free G-graded algebra F hx1,g1 , . . . , xr,gr i, where the xi,gi ’s are non-commuting variables which are in one to one correspondence with the entries of α. As above, we may abuse notation by deleting the index i and simply write xgi . Next we consider the T -ideal of G-graded identities idG (F G) of the group algebra F G. Recall from [2] that idG (F G) is generated as a T -ideal by binomial identities of the form xgi1 xgi2 · · · xgin − xgiσ(1) xgiσ(2) · · · xgiσ(n) where σ is a permutation in Sn and the products gi1 gi2 · · · gin and giσ(1) giσ(2) · · · giσ(n) coincide in G. That is two monomials are equivalent if and only if they have the same variables and they determine elements in the same g-homogeneous component. In particular if the group G is abelian, then two monomials are equivalent if and only if they have the same variables.

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Remark 3.1. Clearly, by passing to a subgroup of G if necessary, we may assume the elements of α = (g1 , g2 , ..., gr ) generate the group G. Fix a natural number n and consider monomials of degree n with n1 variables xg1 , n2 variables xg2 , . . . , nr variables xgr where n1 +n2 +· · ·+nr = n. Now consider permutations of any monomial of that form. Clearly, any permutation determines the same element in the abelianization of G. In other words the elements in G determined by two monomials which have the same variables lie in the same coset ′ of the commutator G . In the next lemma we show that if the monomial is “rich ′ enough” we may obtain all elements of a G -coset. In order to state the lemma we need the following notation. We say that the word Σ = gi1 gi2 · · · gin is a presentation of g ∈ G (in terms of the entries of α = (g1 , g2 , ..., gr )) if g = gi1 gi2 · · · gin in G. For any word Σ we may consider the corresponding monomial (in the free algebra) XΣ = xgi1 xgi2 · · · xgin . We say that the monomial XΣ represents g in G. Lemma 3.2. For every z ∈ G there exists an integer n and a word gi1 gi2 · · · gin in G such that the set of words in Ωgi1 gi2 ···gin = {gσ = giσ(1) giσ(2) · · · giσ(n) , σ ∈ Sym(n)} ′

′

represent all elements of the form zg where g ∈ G (i.e. the full coset of G in G represented by z). Proof. Clearly it is sufficient to find a word whose permutations yield all elements ′ ′ of G . It is well known that the commutator subgroup G of G is generated by commutators [g, h] = ghg −1 h−1 where g, h ∈ G. For any commutator [g, h] we write the elements g and h as words in the entries of α and then we write g −1 and h−1 by inverting the corresponding words. We denote by Σ[g,h] the corresponding word in the entries of α and their inverses (which is equal to [g, h] in G). Clearly, the total degree (in Σ[g,h] ) of each entry gi is zero and hence permuting the elements of Σ[g,h] we obtain the identity element e. It follows easily that taking products of ′ such words we obtain a word Σ = Σz whose different permutations yield all of G . This would complete the proof of the lemma if we assume that whenever g is an entry in α then also g −1 is. But clearly, if this is not the case, we can replace g −1 by g ord(g)−1 and the result follows. Consider the (r-dimensional) lattice Γr = (Z+ )(r) of non negative integer points, where r is the cardinality of α. We refer to Γr as the r-dimensional non-negative Euclidean lattice. Similarly we will consider Γk , k ≤ r, the k-dimensional non→ → negative Euclidean lattices and their translations, − x + Γk where − x ∈ (Z+ )(k) . We view the lattice Γr , as a partial ordered set where A = (n1 , . . . , nr ) ≺ B = (m1 , . . . , mr ) if and only if ni ≤ mi for 1 ≤ i ≤ r. Clearly, this partial order inherits a partial order on Γk , k ≤ r, and their translations. To any point A = (n1 , . . . , nr ), ni ≥ 0 in Γr we attach all monomials XΣ with number of variables as prescribed by the point A, that is, the variable x1,g1 appears n1 times, x2,g2 appears n2 and so on. Clearly any word Σ determines a unique lattice point A in which case we write A = AΣ or Σ ∈ A. Clearly the elements in G represented by all ′ monomials that correspond to a point A ∈ Γr lie in the same coset of G and hence


denoting by NA = {g ∈ G : g is represented by monomials in A}, we have that ′ 1 ≤ ord(NA ) ≤ ord(G ). ′

Lemma 3.3. The function ord(N (A)) : Γr → {1, . . . , ord(G )} is monotonic (increasing) with respect to partial ordering on Γr . In particular if AΣ ∈ Γr , where Σ is a word in the entries of α whose different ′ permutations represent all elements of G (as constructed in Lemma 3.2)), then for ′ any word Π such that AΠ AΣ we have ord(N (AΠ )) = ord(G ). Proof. This is clear.

We can now complete the proof that the Hilbert series of the corresponding relatively free algebra is rational. Let A = (n1 , . . . , nr ) ∈ Γr , where n = n1 +· · ·+nr , and PA be the space spanned by all monomials XΠ where Π ∈ A. Since the T -ideal of identities is spanned by strongly homogeneous polynomials we have that the subspace of the relatively free algebra spanned by monomials of degree n is decomposed into the direct sum of spaces which are spanned by monomials in lattice points A of degree n. ′ We claim that for any integer λ, 1 ≤ λ ≤ ord(G ), the set of points A ∈ Γr such that dim(PA /PA ∩ (idG (F G))) = λ is a finite union of disjoint sets which are translations of lattices Γk , 0 ≤ k ≤ r. We present here a proof which was shown to us by Uri Bader. Consider the one point c compactification Zc + of Z+ . It is convenient to view the space Z+ as homeomorphic to the set of points IN = {1/n : n ∈ N} ∪ {0} with the induced topology of the Lesbegue measure on the interval [0, 1]. The closed sets are either the finite sets or those that contain 0. Consequently the sets which are closed and open are either the finite sets without 0 or sets that contain a set of the form {1/n : n ≥ d} ∪ {0}. Consider the r-fold cartesian product INr = IN × IN × · · · × IN Clearly INr is compact. Furthermore the function ord(N (A)) with values in the ′ finite set T = {1, . . . , ord(G )} (viewed as a function on INr ) is monotonic decreasing and hence continuous. It follows that the inverse image of any point in T is an open and closed subset of INr and the result now follows easily. Returning to Γr = (Z+ )(r) we see that the rationality of the series will follow if we know the rationality of the Hilbert series which corresponds to the enumeration of lattice points of degree n in Γk , k ≤ r. Thus we need to check the rationality of the following power series X ((n + k − 1)!/(n!)(k − 1)!)tn n

and this is clear. This gives a direct proof of Theorem 1.2 in case I = idG (F G). We close the article with an estimate of the sequence of codimensions which corresponds to the T -ideal I = idG (F G). For the calculation we consider the F -space Pn spanned by all G-graded multilinear monomials of degree n on the variables

12


{xg,i }g∈G,i=1,...,n which are permutations of monomials of the form x1,g1 x2,g2 . . . xn,gn , where (g1 , . . . , gn ) ∈ G(n) . Clearly, dim(Pn ) = ord(G)n × n!. Let cnG (F G) = Pn /(Pn ∩ I). The integer cnG (F G) is the n-codimension which corresponds to the G-graded T -ideal I. Corollary 3.4. Let A = F G be the group algebra over a field F and let idG (F G) be the G-graded T -ideal of identities. Let cnG (F G) be the n-th coefficient of the codimension sequence of idG (F G). Then for any integer n we have (1) ′

ord(G)n ≤ cnG (F G) ≤ ord(G ) ord(G)n . (2) ′

lim (cnG (F G)/ ord(G ) ord(G)n )) = 1 p In particular expG (F G) = limn→∞ n cnG (F G) = ord(G) (see [1]). n→∞

Proof. Let x = x1,g1 x2,g2 . . . xn,gn be a G-graded monomial in Pn . As noted above, ′ the set of all n! permutations of x decomposes into at most G equivalence classes where two monomials are equivalent if and only if the difference is a binomial ′ identity of F G. It then follows easily that ord(G)n ≤ cnG (F G) ≤ ord(G ) ord(G)n . For the second part, recall from the second part of Lemma 3.3 that if a monomial contains G-graded variables with degrees in G as in the word Σ (including ′ multiplicities), then its permutations yield precisely G non equivalent classes. So the second part of the corollary will follow easily if we show that lim

n→∞

dn =0 ord(G)n × n!

where dn denotes the number of monomials in Pn which do not contain the set of elements of Σ (with repetitions). But this of course follows from an easy calculation which is omitted. Remark 3.5. One may replace the group algebra F G above by any twisted group algebra F α G, where α is a 2-cocycle of G with values in F ∗ . As above, also here, two monomials are equivalent if and only if they have the same variables and they determine elements in the same g-homogeneous component. The only difference (comparing to the case where α ≡ 1) is that here, for two such monomials, the polynomial identity they determine has the form xgi1 xgi2 · · · xgin − γxgiσ(1) xgiσ(2) · · · xgiσ(n) where γ is a non zero element of F which is determined by the 2-cocycle α and the words gi1 gi2 · · · gin and giσ(1) giσ(1) · · · giσ(1) (see [2]). One sees easily that the twisting by the 2-cocycle α has no effect neither on the Hilbert series nor on the sequence of codimnesions. Details are left to the reader.


References [1] E. Aljadeff, A. Giambruno, D. La Mattina Graded polynomial identities and exponential growth, to appear in J. Reine Angew. Math., arXiv:0903.1860 [2] E. Aljadeff, D. Haile and M. Natapov, Graded identities of matrix algebras and the universal graded algebra, to appear in Trans. Amer. Math. Soc., arXiv:0710.5568. [3] E. Aljadeff and A. Kanel-Belov, Representability and Specht problem for G-graded algebras, Adv. in Math. 225 (2010), pp. 2391-2428. [4] S. A. Amitsur, A generalization of Hilberts Nullstellensatz. Proc. Amer. Math. Soc. 8 (1957), 649656. [5] D.J. Anick, Noncommutative graded algebras and their Hilbert series. J. Algebra 78 (1982), no. 1, 120140. [6] Yu. A. Bahturin, S. K. Sehgal and M. V. Zaicev, Finite-dimensional simple graded algebras, Sb. Math. 199 (2008), no. 7, 965-983. [7] J. P. Bell, The Hilbert series of prime PI rings, Israel J. Math. 139 (2004), 110. [8] A. Ya. Belov, Rationality of Hilbert series with respect to free algebras. (Russian) Uspekhi Mat. Nauk 52 (1997), no. 2(314), 153–154; translation in Russian Math. Surveys 52 (1997), no. 2, 394395. [9] A. Belov, V. Borisenko and V. Latishev Monomial algebras. NY, Plenum. 1998, p. 5 – 190. J.Math.Sci., NY, 87, 1997, no3, pp.3463–3575 [10] A. Berele, Applications of Belov’s theorem to the cocharacter sequence of p.i. algebras, J. Algebra 298 (2006), no. 1, 208–214. [11] A. Berele and A. Regev, Asymptotic behaviour of codimensions of p.i. algebras satisfying Capelli identities , Trans. Amer. Math. Soc. 360 (2008), 5155-5172. [12] A. Braun, The nilpotency of the radical in a finitely generated PI ring. J. Algebra 89 (1984), no. 2, 375396. [13] V. S. Drensky, On the Hilbert series of relatively free algebras. Comm. Algebra 12 (1984), no. 19-20, 23352347. [14] V. S. Drensky, Codimensions of T-ideals and Hilbert series of relatively free algebras. C. R. Acad. Bulgare Sci. 34 (1981), no. 9, 12011204. [15] V. S. Drensky, Codimensions of T-ideals and Hilbert series of relatively free algebras. J. Algebra 91 (1984), no. 1, 117. [16] V.O. Ferreira and L.S.I. Murakami, On free associative algebras linearly graded by finite groups, Papers from the XVIII Latin-American Algebra Colloquium, held in Sao Pedro, Brazil, August 2009. Contemporary Mathematics, American Mathematical Society, Providence, RI, to appear. [17] A. Giambruno and M. Zaicev, On codimension growth of finitely generated associative algebras, Adv. Math. 140 (1998), 145–155. [18] A. Giambruno and M. Zaicev, Exponential codimension growth of P.I. algebras: an exact estimate, Adv. in Math. 142 (1999), 221–243. [19] A. Giambruno and M. Zaicev, Polynomial identities and asymptotic methods. Mathematical Surveys and Monographs, 122. American Mathematical Society, Providence, RI, 2005. [20] A. Kanel-Belov and L.H. Rowen, Computational Aspects of Polynomial Identities, A. K. Peters Ltd., Wellesley, MA. (2005). [21] A. Kanel-Belov and L.H. Rowen Perspectives on Shirshov’s Height Theorem, selected papers of A.I.Shirshov, Birkh¨ user Verlag AG (2009), 3–20. [22] A. R. Kemer, Capelli identities and nilpotency of the radical of finitely generated PI-algebra. (Russian) Dokl. Akad. Nauk SSSR 255 (1980), no. 4, 793797. [23] A. R. Kemer, Ideals of Identities of Associative Algebras, Amer. Math. Soc., Translations of Monographs 87 (1991). [24] A. R. Kemer, Identities of finitely generated algebras over an infinite field (Russian) Izv. Akad. Nauk SSSR Ser. Mat. 54 (1990), no. 4, 726753; English translation in Math. USSRIzv. 37 (1991), no. 1, 6996. [25] Y. Kobayashi, The Hilbert series of some graded algebras and the Poincare series of some local rings. Math. Scand. 42 (1978), no. 1, 1933. [26] P. Koshlukov, Rational Hilbert series of relatively free Jordan algebras. J. Algebra 121 (1989), no. 2, 301309.

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[27] Ju. P. Razmyslov, The Jacobson Radical in PI-algebras. (Russian) Algebra i Logika 13 (1974), 337360, 365. [28] J. B. Shearer, A graded algebra with a nonrational Hilbert series. J. Algebra 62 (1980), no. 1, 228231. [29] R. P. Stanley, Hilbert Functions of Graded Algebras, Adv. Math., 28, (1978), 5783. [30] R. P. Stanley, Combinatorics and commutative algebra. Progress in Mathematics, 41. Birkhuser Boston, Inc., Boston, MA, 1983. [31] I. Sviridova, Identities of PI-algebras graded by a finite abelian group, to appear in Comm. Algebra. Department of Mathematics, Technion-Israel Institute of Technology, Haifa 32000, Israel E-mail address: [email protected] Department of Mathematics, Bar-Ilan University, Ramat-Gan, Israel E-mail address: [email protected]